---

Claim. Corollary 4: $\sum_{n\ge1}1/F_{2^n+1}$ is irrational, so the instance $n_k=2^k+1$ of Problem 267, whose index ratios are at least $5/3$, has answer yes. The paper's main Theorem states that if $(a_n)$ and $(b_n)$ are sequences of positive integers with $a_{n+1}>(b_{n+1}/b_n)a_n^2-(b_{n+1}/b_n)a_n+1$ for all large $n$, then $\sum b_n/a_n$ is irrational; the proof writes the sum as a limit of rationals $A_n/P_n$ with $P_n=a_1\cdots a_n$ and applies Brun's criterion for irrationality of limits of increasing rational sequences. Corollary 4 verifies the criterion for $a_n=F_{2^n+1}$ and $b_n=1$ through the Fibonacci identity $F_{2k+1}=F_k^2+F_{k+1}^2$, which gives $F_{2^{n+1}+1}\ge F_{2^n+1}^2$. Section 5 of the paper answers two questions of Erdős and Graham in this way, the second being Corollary 5, the irrationality of $\sum_{n\ge1}1/L_{2^n}$ over the Lucas numbers, which lies outside this problem. The source is C. Badea, The irrationality of certain infinite series, Glasgow Math. J. 29 (1987), no. 2, 221–228, on the card badea_1987_irrationality_certain_infinite_series.

Covers. The instance $n_k=2^k+1$: the answer is yes. Not covered: every other index sequence. The instance meets the condition of Badea's 1993 corollary with equality, although its ratios are below $2$ (the Badea page).

Acceptance. Refereed: Glasgow Mathematical Journal, volume 29, issue 2 (July 1987). The site's commentary credits Badea with the irrationality of this sum but labels the problem OPEN, so that commentary is not listed as reviewed evidence. The corpus has not reproved the theorem and awards no tier of its own.

Depends on. Nothing in this wiki; the claim rests on the cited paper.